Think of this example:
Collect a dataset based on the coins in peoples pockets, the y variable/response is the total value of the coins, the variable x1 is the total number of coins and x2 is the number of coins that are not quarters (or whatever the largest value of the common coins are for the local).
It is easy to see that the regression with either x1 or x2 would give a positive slope, but when incuding both in the model the slope on x2 would go negative since increasing the number of smaller coins without increasing the total number of coins would mean replacing large coins with smaller ones and reducing the overall value (y).
The same thing can happen any time you have correlalted x variables, the signs can easily be opposite between when a term is by itself and in the presence of others.
Question 1
If your outcome variable is integrated, you might consider using a single-equation generalized error correction model (GECM) as per Banerjee (1993) and De Boef (2001), as this model is agnostic to the stationarity of the predictors.
You might evaluate the stationarity of your outcome using:
$\log{(GDP/Labor)_{ti}} \sim \rho_{i}\log{(GDP/Labor)_{t-1i}} + \zeta_{ti} + \mu_{\rho_{i}}$,
where:
$\zeta_{ti}$ measures all disturbances to $\log{(GDP/Labor)_{ti}}$ in each time $t$ (assumed distributed normal), and
$\mu_{\rho_{i}}$ measures state-level variation in $\log{(GDP/Labor)_{ti}}$ (assumed distributed normal).
If $|\rho_{i}| \approx 1$, then you've got nearly integrated data, and the GECM, which also has the attractive properties of disentangling long-run effects, from both instantaneous change short term effects and from lagged short term effects.
The general form of the single equation GECM is:
$\Delta y_{t} = \beta_{0} + \beta_{c}\left[y_{t-1}-\left(\mathbf{X}_{t-1}\right)\right] + \mathbf{B}_{\Delta\mathbf{X}}\Delta\mathbf{X}_{t} + \mathbf{B}_{\mathbf{X}}\mathbf{X}_{t-1} + \varepsilon$,
where:
$\Delta$ is the first difference operator (e.g. $\Delta y_{t} = y_{t} - y_{t-1}$), and $\varepsilon$ may be decomposed into mixed effects (e.g. by including $\beta_{0i}$, for country-level random intercepts).
instantaneous short run effects are given by $\beta_{\Delta\mathbf{X}}$,
lagged short run effects are given by $\beta_{\mathbf{X}} - \beta_{c} - \beta_{\Delta\mathbf{X}}$, and
long run effects are given by $\left(\beta-{c}-\beta_{\mathbf{X}}\right)/\beta_{c}$.
This specification assumes a homogeneity of error correction processes. I haven't yet tried to derive a heterogeneous error correction specification...
In Stata you can perform Hadri's test for unit-root in panel data on the residuals of such a model, to check them for stationarity.
Question 2
I do not know that I can say much useful here.
Question 3
The time dummies can be included in the GECM model, and presumably other dynamic times series models, often they are used as indicators of, for example, policies going into effect. I have done something similar, but used (time-varying) proportions (rather than 0/1 indicator variables) to represent the portion of the time period during which a policy was in effect (e.g. some policies go into effect January 1, some July 1, some December 21, etc.). On the other hand: you don't have tons of data, so I suppose it depends how many new variables you are adding.
References:
Banerjee, A., Dolado, J. J., Galbraith, J. W., and Hendry, D. F. (1993). Co-integration, error correction, and the econometric analysis of non-stationary data. Oxford University Press, USA.
De Boef, S. (2001). Modeling equilibrium relationships: Error correction models with strongly autoregressive data. Political Analysis, 9(1):78–94.
Best Answer
\begin{aligned} y_i &= \beta_1D1_i + \beta_2D2_i +e_i\\ &=\beta_2 + (\beta_1-\beta_2)D1_i+e_i \end{aligned}
In this sense, you will be able to uniquely estimate $\beta_2$ and $\beta_1 - \beta_2$ and hence $\beta_1$. You won't be able to do this if you specify a intercept term.
For a more formal discussion, you can google Estimability. Because I think your question is more about estimability, instead of multicolinearity.